Bayesian Machine Learning - Lecture 6

Transcription

1 Bayesian Machine Learning - Lecture 6 Guido Sanguinetti Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh gsanguin@inf.ed.ac.uk March 2, 2015

2 Today s lecture 1 Structure and conditional independence 2 3

3 Product rule - revisited Consider a joint probability distribution over random variables x 1,..., x N p(x 1,..., x N ) As probability distributions are symmetric, picking an ordering of the variables is arbitrary Given an arbitrary ordering, I can use the product rule to rewrite the joint as p(x 1,..., x N ) = p(x N x N 1,..., x 1 )p(x N 1 x N 2,..., x 1 )... p(x 1 ) (1) Think of linking all random variables with segments and then stretching the resulting graph Useful to draw samples, but does not encode any deeper structure

4 Factorisations and graphs The product rule gives us a generic but rather useless factorisation of joint probabilities Interesting things arise when special factorisations are present, i.e. when you can remove some variables from the conditioning sets in (1) Today we will formalise what interesting means by mapping distributions onto graphs Graphs are also a very useful tool to model systems, as they abstract the dependence structure between variables without bothering with the details of the specific distributions

5 Conditional independence Two variables x 1 and x 2 are conditionally independent given x 3 (x 1 x 2 x 3 ) if Equivalently p (x 1 x 2, x 3 ) = p (x 1 x 3 ). p(x 1, x 2 x 3 ) = p(x 1 x 3 )p(x 2 x 3 ) Two non-overlapping sets of nodes A and B are d-separated by a set C if every node in A is conditionally independent of every other node in B given the nodes in C Given a variable (node) x i, its Markov blanket is the set of nodes M that d-separate x i from the rest of the network

6 DAGs and distributions Let us assume we have a factorisation of joint probability distribution in terms of conditionals A factorization defines a Directed Acyclic Graph (DAG) in the following way: variables are associated with nodes, and variables to the right of the conditioning sign (parents) are linked with an arrow to variables to the left (children) Warning: the graph is not unique (Markov equivalence)!!! Cool thing, start with a DAG and get a distribution with structure

7 Graphical notation Random variables are denoted by circles with the name inside Observed random variables are often shaded I.i. d. variables are denoted through plates Sometimes, additional parameters which we are not doing posterior inference over are denoted as squares Let s draw the graphical model for the Gaussian Mixture Model

8 Francesco s example Francesco works on modelling users from twitter data. Let s draw a graphical model for him!

9 Working out conditional independences from graphs Graphically, two variables a and b are d-separated by C if every path between a and b is blocked A path is blocked if the arrows meet head to tail or tail to tail at a node in C A path is blocked also if there exists a node x on the path where arrows meet head to head, and neither x nor its descendants are in C The Markov blanket of a node are its children, parents and co-parents Notice that not all conditional independence can be represented by a DAG (rather pathological counter examples)

10 Dependencies with no direction Sometimes we wish to encode dependencies which do not have an obvious precedence Discuss some examples In these cases, we just draw lines between nodes without arrows Hybrid examples (some directed edges, some undirected) also possible One particular example from my work (Sanguinetti et al, Bioinformatics 2008)

11 Conditional independence and Markov Random Fields Formally, we can associate an undirected graphical model (Markov Random Field) to a general factorisation of a joint distribution (not necessarily in terms of conditional distributions) Nodes are connected if they appear together in a factor Maximal cliques in the graph correspond to factors Conditional independence is equivalent to separation in the graph (easy!)

12 Simple example Consider the linear dynamical system x t = ax t 1 + ɛ t ɛ t N (0, σ 2 ) x 0 N (0, 1) (2) Write the joint distribution Draw the corresponding directed and undirected graphical models

13 From directed to undirected graphs In general, it is possible to go from a directed to an undirected graphical model (why?) Remove all arrows Join all nodes that share a child (moralisation) Are conditional independences preserved?